Minimum Spanning Tree (MST) is a fundamental structure in graph analytics and can be applied in various applications. The problem of maintaining MSTs in dynamic graphs is significant, as many real-world graphs are frequently updated. Existing studies on MST maintenance primarily focus on theoretical analysis and lack practical efficiency. In this paper, we propose a novel algorithm to maintain MST in dynamic graphs, which achieves high practical efficiency. In addition to the tree structure, our main idea is to maintain a replacement edge for each tree edge. In this way, the tree structure can be immediately updated when a tree edge is deleted. We propose algorithms to maintain the replacement edge for each tree edge by sharing the computation cost in the updating process. Our performance studies on large datasets demonstrate considerable improvements over state-of-the-art solutions.

We study the historical connectivity query in temporal graphs where edges continuously arrive. Given an arbitrary time window, and two query vertices, the problem aims to identify if two vertices are connected by a path in the snapshot of the window. The state-of-the-art method designs an index based on the two-hop cover, and updating the index is costly when new edges arrive. In this paper, we propose a new framework and design a novel forest-based index for historical connectivity queries. The index enables us to answer queries by searching if two vertices are connected in the forest. We update the index by modifying a forest structure. Our techniques also work for connectivity query processing in a sliding window of temporal graphs. Extensive experiments have been conducted to show the considerable advantages of our approach compared with the state-of-the-art methods in both historical connectivity queries and sliding-window connectivity queries.

Connectivity query processing is a fundamental problem in graph processing. Given an undirected graph and two query vertices, the problem aims to identify whether they are connected via a path. Given frequent edge updates in real graph applications, in this paper, we study connectivity query processing in fully dynamic graphs, where edges are frequently inserted or deleted. A recent solution, called D-tree, maintains a spanning tree for each connected component and applies several heuristics to reduce the depth of the tree. To improve the efficiency, we propose a new spanning-tree-based solution by maintaining a disjoint-set tree simultaneously. By combining the advantages of two trees, we achieve the constant query time complexity and also significantly improve the theoretical running time in both edge insertion and edge deletion. Our performance studies on real large datasets show considerable improvement of our algorithms.